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Theorem smgrpisass 21016
Description: A semi-group is associative. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
Assertion
Ref Expression
smgrpisass  |-  ( G  e.  SemiGrp  ->  G  e.  Ass )

Proof of Theorem smgrpisass
StepHypRef Expression
1 elin 3371 . . 3  |-  ( G  e.  ( Magma  i^i  Ass ) 
<->  ( G  e.  Magma  /\  G  e.  Ass )
)
21simprbi 450 . 2  |-  ( G  e.  ( Magma  i^i  Ass )  ->  G  e.  Ass )
3 df-sgr 21014 . 2  |-  SemiGrp  =  (
Magma  i^i  Ass )
42, 3eleq2s 2388 1  |-  ( G  e.  SemiGrp  ->  G  e.  Ass )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696    i^i cin 3164   Asscass 20995   Magmacmagm 21001   SemiGrpcsem 21013
This theorem is referenced by:  mndoisass  25459
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172  df-sgr 21014
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